The Hele-Shaw flow and moduli of holomorphic discs
Abstract
We present a new connection between the Hele-Shaw flow, also known as two-dimensional (2D) Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this we prove short time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also starting from a smooth Jordan domain. Applying the same ideas we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.
Keywords
Cite
@article{arxiv.1212.2337,
title = {The Hele-Shaw flow and moduli of holomorphic discs},
author = {Julius Ross and David Witt Nystrom},
journal= {arXiv preprint arXiv:1212.2337},
year = {2015}
}
Comments
27 pages, 3 figures. v2 has improved exposition throughout, and some small correctifications (in particular to the proof of the short time existence with empty initial condition)